Steps in Differential Geometry, Proceedings of the Colloquium on Differential Geometry, 25–30 July, 2000, Debrecen, Hungary HAMILTONIAN FIELD THEORY REVISITED: A GEOMETRIC APPROACH TO REGULARITY OLGA KRUPKOVA´ Abstract. The aim of the paper is to announce some recent results con- cerning Hamiltonian theory for higher order variational problems on fibered manifolds. A reformulation, generalization and extension of basic concepts such as Hamiltonian system, Hamilton equations, regularity, and Legendre transformation, is presented. The theory is based on the concept of Lepagean (n + 1)-form (where n is the dimension of the base manifold). Contrary to the classical approach, where Hamiltonian theory is related to a single Lagrangian, within the present setting a Hamiltonian system is associated with an Euler– Lagrange form, i.e., with the class of all equivalent Lagrangians. Hamilton equations are introduced to be equations for integral sections of an exterior differential system, defined by a Lepagean (n + 1)-form. Relations between extremals and solutions of Hamilton equations are studied in detail. A revi- sion of the concepts of regularity and Legendre transformation is proposed, reflecting geometric properties of the related exterior differential system. The new look is shown to lead to new regularity conditions and Legendre trans- formation formulas, and provides a procedure of regularization of variational problems. Relations to standard Hamilton–De Donder theory, as well as to multisymplectic geometry are studied. Examples of physically interesting La- grangian systems which are traditionally singular, but regular in this revised sense, are discussed. 1. Introduction Hamiltonian theory belongs to the most important parts of the calculus of vari- ations. The idea goes back to the first half of the 19th century and is due to Sir William Rowan Hamilton and Carl Gustav Jacob Jacobi who, for the case of clas- sical mechanics, developed a method to pass from the Euler–Lagrange equations to another set of differential equations, now called Hamilton equations, which are “better adapted” to integration. This celebrated procedure, however, is applicable 1991 Mathematics Subject Classification. 35A15, 49L10, 49N60, 58Z05. Key words and phrases. Lagrangian system, Poincar´e-Cartan form, Lepagean form, Hamil- tonian system, Hamilton extremals, Hamilton–De Donder theory, Hamilton equations, regularity, Legendre transformation. Research supported by Grants MSM:J10/98:192400002 and VS 96003 of the Czech Ministry of Education, Youth and Sports, and GACR 201/00/0724 of the Czech Grant Agency. The author also wishes to thank Professors L. Kozma and P. Nagy for kind hospitality during the Colloquium on Differential Geometry, Debrecen, July 2000. 187 188 OLGA KRUPKOVA´ only to a certain class of variational problems, called regular. Later the method was formally generalized to higer order mechanics, and both first and higher order field theory, and became one of the constituent parts of the classical variational theory (cf. [8], [4]). In spite of this fact, it has been clear that this generaliza- tion of Hamiltonian theory suffers from a principal defect: allmost all physically interesting field Lagrangians (gravity, Dirac field, electromagnetic field, etc.) are non-regular, hence they cannot be treated within this approach. Since the second half of the 20’th century, together with an increasing interest to bring the more or less heuristic classical variational theory to a modern framework of differential geometry, an urgent need to understand the geometric meaning of the Hamiltonian theory has been felt, in order to develop its proper generalizations as well as global aspects. There appeared many papers dealing with this task in different ways, with results which are in no means complete: from the most important ones let us mention here at least Goldschmidt and Sternberg [14], Aldaya and Azc´arraga [1], Dedecker [5], [7], Shadwick [37], Krupka [21]–[23], Ferraris and Francaviglia [9], Krupka and Stˇep´ankov´a[26],ˇ Gotay [16], Garcia and Mu˜noz [10], [11], together with a rather pessimistic Dedecker’s paper [6] summarizing main problems and predicting that a way-out should possibly lead through some new understanding of such fundamental concepts as regularity, Legendre transformation, or even the Hamiltonian theory as such. The purpose of this paper is to announce some very recent results, partially pre- sented in [30]–[32] and [38], which, in our opinion, open a new way for understanding the Hamiltonian field theory. We work within the framework of Krupka’s theory of Lagrange structures on fibered manifolds where the so called Lepagean form is a central concept ([18], [19], [21], [24], [25]). Inspired by fresh ideas and inter- esting, but, unfortunately, not very wide-spread “nonclassical” results of Dedecker [5] and Krupka and Stˇep´ankov´a[26],ˇ the present geometric setting means a direct “field generalization” of the corresponding approach to higher order Hamiltonian mechanics as developed in [27] and [28] (see [29] for review). The key point is the concept of a Hamiltonian system, which, contrary to the usual approach, is not re- lated with a single Lagrangian, but rather with an Euler–Lagrange form (i.e., with the class of equivalent Lagrangians), as well as of regularity, which is understood to be a geometric property of Hamilton equations. It turns out that “classical” results are incorporated as a special case in this scheme. Moreover, for many variational systems which appear singular within the standard approach, one obtains here a regular Hamiltonian counterpart (Hamiltonian, independent momenta which can be considered a part of certain Legendre coordinates, Hamilton equations equivalent with the Euler–Lagrange equations). This concerns, among others, such important physical systems as, eg., gravity, electromagnetism or the Dirac field, mentioned above. HAMILTONIAN FIELD THEORY REVISITED 189 2. Notations and preliminaries All manifolds and mappings throughout the paper are smooth. We use standard notations as, eg., T for the tangent functor, J r for the r-jet prolongation functor, d for the exterior derivative of differential forms, iξ for the contraction by a vector field ξ, and ∗ for the pull-back. We consider a fibered manifold (i.e., surjective submersion) π : Y → X, dim X = r n, dim Y = m + n, its r-jet prolongation πr : J Y → X, r ≥ 1, and canonical jet r k 0 projections πr,k : J Y → J Y , 0 ≤ k < r (with an obvious notation J Y = Y ). A fibered chart on Y is denoted by (V, ψ), ψ = (xi, yσ), the associated chart on J rY by (V , ψ ), ψ = (xi, yσ, yσ , . , yσ ). r r r j1 j1...jr r A vector field ξ on J Y is called πr-vertical (respectively, πr,k-vertical) if it k projects onto the zero vector field on X (respectively, on J Y ). We denote by V πr r the distribution on J Y spanned by the πr-vertical vector fields. r k A q-form ρ on J Y is called πr,k-projectable if there is a q-form ρ0 on J Y such ∗ r that πr,kρ0 = ρ.A q-form ρ on J Y is called πr-horizontal (respectively, πr,k- horizontal) if iξρ = 0 for every πr-vertical (respectively, πr,k-vertical) vector field ξ on J rY . The fibered structure of Y induces a morphism, h, of exterior algebras, defined by the condition J rγ∗ρ = J r+1γ∗hρ for every section γ of π, and called the hor- izontalization. Apparently, horizontalization applied to a function, f, and to the elements of the canonical basis of 1-forms, (dxi, dyσ, dyσ , . , dyσ ), on J rY j1 j1...jr gives hf = f ◦ π , hdxi = dxi, hdyσ = yσdxl, . , hdyσ = yσ dxl. r+1,r l j1...jr j1...jr l A q-form ρ on J rY is called contact if hρ = 0. On J rY , behind the canonical basis of 1-forms, we have also the basis (dxi, ωσ, ωσ , . , ωσ , dyσ ) adapted j1 j1...jr−1 j1...jr to the contact structure, where in place of the dy’s one has the contact 1-forms ωσ = dyσ − yσdxl, . , ωσ = dyσ − yσ dxl. l j1...jr−1 j1...jr−1 j1...jr−1l Sections of J rY which are integral sections of the contact ideal are called holonomic. Apparently, a section δ : U → J rY is holonomic if and only if δ = J rγ where γ : U → Y is a section of π. Notice that every p-form on J rY , p > n, is contact. Let q > 1. A contact r r q-form ρ on J Y is called 1-contact if for every πr-vertical vector field ξ on J Y the (q − 1)-form iξρ is horizontal. Recurrently, ρ is called i-contact, 2 ≤ i ≤ q, if r iξρ is (i − 1)-contact. Every q-form on J Y admits a unique decomposition ∗ πr+1,rρ = hρ + p1ρ + p2ρ + ··· + pqρ, r+1 where piρ, 1 ≤ i ≤ q, is an i-contact form on J Y , called the i-contact part of ρ. It is helpful to notice that the chart expression of piρ in any fibered chart contains exactly i exterior factors ωσ where l is admitted to run from 0 to r. For more j1...jl details on jet prolongations of fibered manifolds, and the calculus of horizontal and contact forms the reader can consult eg. [18], [19], [24], [25], [29], [33], [34]. 190 OLGA KRUPKOVA´ Finally, throughout the paper the following notation is used: 1 2 n ω0 = dx ∧ dx ∧ ... ∧ dx , ωi = i∂/∂xi ω0, ωij = i∂/∂xj ωi, etc. 3. Hamiltonian systems In this section we discuss the concept of a Hamiltonian system and of a La- grangian system as introduced in [30], and the relation between Hamiltonian and Lagrangian systems. Let s ≥ 0, and put n = dim X.A closed (n + 1)-form α on J sY is called a Lepagean (n + 1)-form if p1α is πs+1,0-horizontal.